Part of the Atlas of Small Regular Polytopes

Polytope of Type {30,6}

Atlas Canonical Name {30,6}*1800b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1800,575)
Rank
3
Schläfli Type
{30,6}
Vertices, edges, …
150, 450, 30
Order of s0s1s2
6
Order of s0s1s2s1
30
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

6-fold

25-fold

50-fold

75-fold

150-fold

225-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s0)^2*s2*s1*s0*(s2*s1)^2> of order 2

15 facets

75 vertex figures

P/N, where N=<(s0*s1)^6> of order 5

14 facets

30 vertex figures

P/N, where N=<s0*s1*s2*(s1*s0)^3*s1*s2*s1> of order 5

6 facets

30 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18)(15,17)(26,51)(27,55)(28,54)(29,53)(30,52)(31,71)(32,75)(33,74)(34,73)(35,72)(36,66)(37,70)(38,69)(39,68)(40,67)(41,61)(42,65)(43,64)(44,63)(45,62)(46,56)(47,60)(48,59)(49,58)(50,57);;
s1 := ( 1,27)( 2,26)( 3,30)( 4,29)( 5,28)( 6,33)( 7,32)( 8,31)( 9,35)(10,34)(11,39)(12,38)(13,37)(14,36)(15,40)(16,45)(17,44)(18,43)(19,42)(20,41)(21,46)(22,50)(23,49)(24,48)(25,47)(51,52)(53,55)(56,58)(59,60)(61,64)(62,63)(66,70)(67,69)(72,75)(73,74);;
s2 := ( 2,25)( 3,19)( 4,13)( 5, 7)( 8,24)( 9,18)(10,12)(14,23)(15,17)(20,22)(26,51)(27,75)(28,69)(29,63)(30,57)(31,56)(32,55)(33,74)(34,68)(35,62)(36,61)(37,60)(38,54)(39,73)(40,67)(41,66)(42,65)(43,59)(44,53)(45,72)(46,71)(47,70)(48,64)(49,58)(50,52);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(75)!( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18)(15,17)(26,51)(27,55)(28,54)(29,53)(30,52)(31,71)(32,75)(33,74)(34,73)(35,72)(36,66)(37,70)(38,69)(39,68)(40,67)(41,61)(42,65)(43,64)(44,63)(45,62)(46,56)(47,60)(48,59)(49,58)(50,57);
s1 := Sym(75)!( 1,27)( 2,26)( 3,30)( 4,29)( 5,28)( 6,33)( 7,32)( 8,31)( 9,35)(10,34)(11,39)(12,38)(13,37)(14,36)(15,40)(16,45)(17,44)(18,43)(19,42)(20,41)(21,46)(22,50)(23,49)(24,48)(25,47)(51,52)(53,55)(56,58)(59,60)(61,64)(62,63)(66,70)(67,69)(72,75)(73,74);
s2 := Sym(75)!( 2,25)( 3,19)( 4,13)( 5, 7)( 8,24)( 9,18)(10,12)(14,23)(15,17)(20,22)(26,51)(27,75)(28,69)(29,63)(30,57)(31,56)(32,55)(33,74)(34,68)(35,62)(36,61)(37,60)(38,54)(39,73)(40,67)(41,66)(42,65)(43,59)(44,53)(45,72)(46,71)(47,70)(48,64)(49,58)(50,52);
poly := sub<Sym(75)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

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